Difference Between Turbulence Models

Because all turbulence models offer advantages and disadvantages over others in certain applications, no single turbulence model can be judged to be the optimum model in all circumstances [6]. Thus, the difference between turbulence models is studied in this chapter. The observed turbulence models are the standardk−ε, Re-Normalization Group (RNG)k−ε, realizable k−ε, standard k−ω, Shear-Stress Transition (SST)k−ω, and Reynolds Stress Model (RSM). The choosing of the turbulence model is important, especially when the Eulerian-Lagrangian approach is used, because the direction of the velocity vectors has an influence on the particle trajectories.

When the fluid changes its direction for example in the pipe bend, it experiences an adverse pressure gradient. Because of the viscous effects, the fluid does not have enough momentum to overcome the adverse pressure gradient and separation occurs. In addition to adverse pressure gradient in the streamwise direction, there is a pressure gradient acting radially because of the centripetal forces. In curved ducts and pipes, secondary flows are mainly generated by these centripetal forces, but they can also be generated by the anisotropy in the Reynolds stresses. [52]

Because secondary flows and boundary layer separation affect the particle trajecto-ries, they are studied in the following in conjunction with the turbulence models.

6.1.1 Secondary Flows

According to Speziale et al. [79], the turbulence structure of flows in the ducts and pipes can be altered significantly by secondary flows. To balance the centrifugal force on the fluid due to its curved trajectory, there must be a pressure gradient across the pipe. This pressure gradient generates the centripetal force, which is equal and opposite to centrifugal force. Due to viscosity, the fluid near the top and bottom walls of the pipe moves more slowly than that in the middle of the pipe

and therefore requires a smaller pressure gradient to balance its centrifugal force.

Consequently, a secondary flow occurs in which the fluid near the top and bottom walls of the pipe moves from the outer side towards the center of the curvature and the fluid in the middle of the pipe moves from the inner side outwards. Due to the secondary flow, the position of the maximum axial flow velocity is shifted towards the outer wall of the pipe from the pipe centerline. [80, 81]

In straight ducts and pipes, secondary flows are not generated by centripetal forces but by the anisotropy in the Reynolds stresses. Also in the curved pipes with weak curvature ratios (the ratio of the radius of the curvature to the pipe radius,R₀), the Reynolds stress differences play a crucial role in generating secondary flows. Con-sequently, two-equation models based on the Boussinesq approximation are inca-pable of predicting secondary flows in straight ducts and pipes and in the ducts and pipes with weak curvature ratios. However, two-equation models have yielded rea-sonably acceptable predictions for fully-developed secondary flows in curved ducts and pipes with moderate to strong curvature ratios. [79]

Also the forcing on the flow by particles distributed non-uniformly in the cross-section can generate secondary flows. In a turbulent flow, particles can generate changes in turbulence, which in the case of non-uniformly distributed particles, re-sults in an anisotropy in the Reynolds stresses in the cross-section, which generates secondary flows. [82]

The strength of the secondary flow is characterized by the Dean number,De, De= Re

R₀, (144)

whereReis the Reynolds number andR₀ the curvature ratio [83]. With increasing Dean number the maximum axial flow velocity moves towards the outer wall of the pipe bend from the pipe centerline [84].

When the velocity fields in the by-pass pipe before (position a in Figure 15) and after (position b in Figure 15) the bend are observed, it can be concluded that there are no significant secondary flows generated by anisotropy in Reynolds stresses

because the results of realizable k−ε and Reynolds stress model are equal and we are observing the pipe with strong curvature ratio (R₀=3). In Figure 15, the flow direction in the by-pass pipe is shown by red arrow, the positions where the secondary flows are observed are shown by green circles, and the effluent pipes are numbered from 1 to 4. The velocity fields before the bend are shown in Figure 16, and after the bend in Figure 17. The only difference between the realizablek−εand Reynolds stress models occurs after the pipe bend where the recirculating region is located. In straight effluent pipes, there is no significant difference between the turbulence models either.

Figure 15.The positions where the secondary flows are observed. a) before the bend b) after the bend

In Figures 16 and 17, the directions of secondary flows generated by centripetal forces in the vicinity of the pipe bend are marked with arrows. Likewise in the study of Jayanti et al. [85], in a 90-degree bend a relatively weak, double-vortex secondary flow is formed, and it grows stronger as the bend angle increases. The flow directs from the inner side to the outer side of the bend in major part of the core, and reverses direction close to the wall. The effect of the secondary flows on particles is studied in Chapter 6.3.

Figure 16.Secondary flow in the cross-section of the by-pass pipe before the bend. a) Realizable k−εb) RSM

Figure 17.Secondary flow in the cross-section of the by-pass pipe after the bend. a) Realizable k−εb) RSM

6.1.2 Boundary Layer Separation

In addition to secondary flows, boundary layer separation has also an influence on particle trajectories. The reattachment length in the case of recirculating region is used here to compare the turbulence models. In the studied geometry, recirculating regions are located for example after the inlets of the duct, after the elbow in the

by-pass pipe, before the opening into the mixing tank, and inside the mixing tank.

Generally, recirculating regions are located where the direction of the main flow changes.

The reattachment length can be found by plotting the values of wall shear stress,τw. The values of wall shear stress are observed along the front wall of the duct, which leads wastewater from the primary settling tank to the mixing tank. The values calculated by six turbulence models are shown in Figure 18. The separation occurs where the wall shear stress gets the first nonzero value at the edge of the inlet. The reattachment occurs when the wall shear stress gets the value of zero.

The separation region can be seen in Figures 19 and 20, where the contours of velocity in the direction of thez coordinate are shown in the left hand side and in the right hand side of the duct, respectively. In Figure 19, the influent flows to the positive direction of thezcoordinate. The separation region is the blue region where the direction of the flow is to the negative direction of thezcoordinate. On the contrary, in Figure 20 the influent flows to the negative direction of thezcoordinate and thus the values of velocity are negative. Now, the separation region is the red region where the direction of the flow is to the positive direction of thezcoordinate.

In both figures, the edge of the inlet where the separation occurs, is marked with black line.

In the case of slurries, it is important that the turbulence is modeled in the most accurate way, because the particles tend to get caught up into the turbulent eddies and recirculating regions. Also the influence of the particles on the flow field should be taken properly into account. Figure 21 shows that all studied turbulence models predict that particles attenuate turbulence, when the particle size is 3.75 mm and particle density 1075 kg/m³. The grey, vertical lines in figure show the values for the dispersed phase volume fraction, which are 1·10⁻⁴ for wastewater from the primary settling tanks and 3·10⁻⁴for overflow wastewater.

However, as discussed in Chapter 4.8, there is no generally accepted model to take

Figure 18.The values of wall shear stress along the front wall of the duct from the primary settling tank calculated by different turbulence models.

Figure 19.The contour plots of velocity in the left hand side of the duct. a) Standardk−εb) RNG k−εc) Realizablek−εd) Standardk−ωe) SSTk−ω f) RSM

Figure 20.The contour plots of velocity in the right hand side of the duct. a) Standardk−εb) RNGk−εc) Realizablek−εd) Standardk−ω e) SSTk−ωf) RSM

the effect of particles on the continuous phase turbulence into account. In ANSYS Fluent, the effect of particles on the continuous phase turbulence can only be taken into account in the discrete phase model. Thus, the attenuation of turbulence by particles cannot be validated in this thesis by studying turbulence modulation with all multiphase models.

It should be noted that Kolmogorov time scale is inversely proportional to turbu-lence dissipation rate. Thus, near the walls where turbuturbu-lence dissipation rate gets larger values, Kolmogorov time scale gets smaller values and if the ratio between particle response time and Kolmogorov time scale gets a value greater than 100, the particles augment turbulence. On the contrary, far from the walls where turbu-lence dissipation rate gets smaller values, Kolmogorov time scale gets larger values and if the ratio between particle response time and Kolmogorov time scale gets a value smaller than 100, the particles attenuate turbulence. The influence of the particles on turbulence is thus dependent on the location with respect to the wall surface. However, when the particles are small enough, they may get trapped in the

Figure 21. The map of regimes of interaction between particles and turbulence. Modified from [54].

viscous sublayer, where the turbulent eddies are not able to pick them to the main stream [56].

In this thesis, there is no experimental data available for comparison. According to literature review, the realizable k−ε and SST k−ω models should the most ac-curately predict the flow with adverse pressure gradients and separation. Results represented in Figures 18-20 show that those models give approximately equal re-sults for the reattachment length. The rere-sults given by the RNGk−ε model differ from the results of other turbulence models. It also slightly underestimates turbu-lence kinetic energy and turbuturbu-lence dissipation rate in the mixing tank. Because of the results and the knowledge that the RNG k−ε model is more appropriate for swirling flows, it is left outside of the following simulation cases. Also the standard k−ε, standardk−ω, and Reynolds Stress model underestimate the reattachment length compared to the realizablek−ε and SSTk−ω models. The standardk−ε and standardk−ω models overestimate turbulence kinetic energy in pipe entrance regions compared to other models as the contour plots of turbulence kinetic energy

in Figure 22 show, but overall there is no significant discrepancies between studied turbulence models.

The realizablek−ε model is preferred to instead of the SSTk−ω model because the computational time of the SSTk−ω model is 1.2 times the computational time used by the realizable k−ε model and the realizable k−ε model also enhances numerical stability. The aim is to achieve converged solution and the available time for simulations is limited. Since there is no experimental data available for com-parison, the realizablek−ε model is chosen for following simulations because it neither overestimates nor underestimates the observed turbulence quantities com-pared to other models.

Figure 22.The contours of turbulence kinetic energy. a) Standardk−εb) RNGk−εc) Realizable k−εd) Standardk−ωe) SSTk−ω f) RSM